Here we seek to overcome the issue of robustness, in many data-driven models, by using the fact that often in dissipative partial differential equations (PDEs), stability comes from a linear operator that dissipates energy [1]}.
In particular, we are concerned with the nonlinear stability of attractors.
By nonlinear stability we mean finite perturbations to states on the attractor are pushed back to the attractor. This type of stability can be show using the energy method [2]}, with Lyapunov functions [3]}, or for dynamical systems which possess an inertial manifold [1]}. The energy method in particular highlights the importance of the linear term in stabilizing many dissipative PDEs. For example, when computing the energy balance for the Viscous Burgers Equation (VBE) and the Navier-Stokes Equation dissipation comes from the diffusive term [5]} and for the Kuramoto-Sivashinky equation (KSE) hyperdiffusivity dissipates energy [6]}. Below we make this notion of stability more precise.
